Question

A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of ${45}^{\circ }$ with the X-axis. The two motions are given by

$x=x_{0}sin\omega t$ and $s=s_{0}sin\omega t$

Find the amplitude of the resultant motion.

• A. $\sqrt{x_0^2+s_0^2+\sqrt{2}{x}_0{s}_0}$
• B. $x_0+s_0$
• C. $\frac{x_0+s_0}{2}$
• D. None of these

Answer 1

The correct option is A

$\sqrt{x_0^2+s_0^2+\sqrt{2}{x}_0{s}_0}$

The particle is subjected to two simple harmonic motions represented by,

$x=x_{0}sin\omega t$

$s=s_{0}sin\omega t$

and, angle between two motions $={45}^{\circ }$

$\therefore$ Resultant motion will be given by,

$R=\sqrt{(x^2+s^2+2xscos{45}^{\circ })}$

$=\sqrt{\{x_0^{2}si{n}^2\omega t+s_0^{2}si{n}^2\omega t+2x_0{s}_{0}si{n}^2\omega t(\frac{1}{\sqrt{2}}\left)\right\}}$

$=[x_0^2+s_0^2+\sqrt{2}{x}_0{s}_0{]}^{\frac{1}{2}}sin\omega t$

$\therefore$ Resultant amplitude